Mon. Not. R. Astron. Soc. 000, 1–12 (2017) Printed 8 September 2017 (MN LATEX style file v2.2)

New constraints on time-dependent variations of fundamentalconstants using Planck data

Luke Hart1? and Jens Chluba1†1Jodrell Bank Centre for Astrophysics, Alan Turing Building, University of Manchester, Manchester M13 9PL

Accepted 2017 –. Received 2017 May 10.

ABSTRACTObservations of the cosmic microwave background (CMB) today allow us to answer detailedquestions about the properties of our Universe, targeting both standard and non-standardphysics. In this paper, we study the effects of varying fundamental constants (i.e., the fine-structure constant, αEM, and electron rest mass, me) around last scattering using the recom-bination codes CosmoRec and Recfast++. We approach the problem in a pedagogical man-ner, illustrating the importance of various effects on the free electron fraction, Thomson vis-ibility function and CMB power spectra, highlighting various degeneracies. We demonstratethat the simpler Recfast++ treatment (based on a three-level atom approach) can be usedto accurately represent the full computation of CosmoRec. We also include explicit time-dependent variations using a phenomenological power-law description. We reproduce pre-vious Planck 2013 results in our analysis. Assuming constant variations relative to the stan-dard values, we find the improved constraints αEM/αEM,0 = 0.9993 ± 0.0025 (CMB only)and me/me,0 = 1.0039 ± 0.0074 (including BAO) using Planck 2015 data. For a redshift-dependent variation, αEM(z) = αEM(z0) [(1 + z)/1100]p with αEM(z0) ≡ αEM,0 at z0 = 1100,we obtain p = 0.0008 ± 0.0025. Allowing simultaneous variations of αEM(z0) and p yieldsαEM(z0)/αEM,0 = 0.9998± 0.0036 and p = 0.0006± 0.0036. We also discuss combined limitson αEM and me. Our analysis shows that existing data is not only sensitive to the value of thefundamental constants around recombination but also its first time derivative. This suggeststhat a wider class of varying fundamental constant models can be probed using the CMB.

Key words: recombination – fundamental physics – cosmology – CMB anisotropies

1 INTRODUCTION

Nowadays, measurements of the cosmic microwave background(CMB) anisotropies allow us to constrain the standard cosmologi-cal parameters with unprecedented precision (Bennett et al. 2013;Planck Collaboration et al. 2015a). This has opened a route for test-ing possible extensions to the ΛCDM model, e.g., related to the ef-fective number of neutrino species and their mass (see Gratton et al.2008; Battye & Moss 2014; Abazajian et al. 2015) and Big BangNucleosynthesis (Coc et al. 2013; Andre et al. 2014; Abazajianet al. 2016). In the analysis, we are furthermore sensitive to percentlevel effects in the recombination dynamics (Rubino-Martın et al.2010a; Shaw & Chluba 2011), which can be captured using ad-vanced recombination codes such as CosmoRec and HyRec (Chluba& Thomas 2011; Ali-Haımoud & Hirata 2011), again emphasizingthe impressive precision of available cosmological datasets.

Our interpretation of the CMB measurements relies on severalassumptions. The validity of general relativity and atomic physicsaround recombination are two evident ones. This encompasses asignificant extrapolation of local physics, tested in the lab, to cos-

? Email: luke.hart@postgrad.manchester.ac.uk† Email: jens.chluba@manchester.ac.uk

mological scales (both in distance and time). Albeit the successesof the ΛCDM cosmology, we know that simple extrapolation is cur-rently not enough to explain the existence of dark matter and darkenergy in our Universe. Similarly, it is important to test the validityof local physical laws in different regimes.

One of these tests is related to the constancy of fundamentalconstants (see Uzan 2003, 2011, for review). This could provide aglimpse at physics beyond the standard model, possibly sheddinglight on the presences of additional scalar fields and their couplingto the standard sector. Variations of the fine-structure constant, αEM,and electron rest mass, me can directly impact CMB observables, asstudied previously (e.g., Kaplinghat et al. 1999; Avelino et al. 2000;Battye et al. 2001; Avelino et al. 2001; Rocha et al. 2004; Martinset al. 2004; Scoccola et al. 2009; Menegoni et al. 2012) using mod-ified versions of recfast (Seager et al. 1999). Similarly, changesof the gravitational constant can be considered (e.g., ??Galli et al.2011). Using Planck 2013 data, the values of αEM and me aroundrecombination were proven to coincide with those obtained in thelab to within ' 0.4% for αEM and ' 1−6% for me (Planck Collabo-ration et al. 2015b). This is ' 2−3 orders of magnitude weaker thanconstraints obtained from ‘local’ measurements (Bize et al. 2003;Rosenband et al. 2008; Bonifacio et al. 2014; Kotus et al. 2017);however, the CMB places limits during very different phases in the

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history of the Universe, some 380, 000 years after the Big Bang,which complement these low-redshift measurements.

In this paper, we describe the effects of varying fundamentalconstants on the cosmological recombination history, focusing onvariations of αEM and me. These directly affect the atomic physicsand radiative transfer in the recombination era (z ' 103) and thuscan be probed using CMB anisotropy measurements. We approachthe problem is a pedagogical manner, illustrating the individual ef-fects on the recombination dynamics in Sec. 2.1. We show that thefull recombination calculation of CosmoRec can be accurately rep-resented using a simple three-level atom approach, by introducingappropriate corrections functions (see Sect. 2.1.1). We discuss con-stant changes of αEM and me, but also introduce a phenomenologi-cal power-law redshift-dependence of these parameters around re-combination. The associated effects on the ionization history aredistinct and thus can be individually constrained using CMB data.

The changes to the recombination codes are then propagatedto the Thomson visibility function and the calculations of the CMBpower spectra. Here we do not focus on the individual contributionsto the CMB power spectrum deviations as these have been coveredin previous literature (see Planck Collaboration et al. 2015b, Ap-pendix B). However, we illustrate the overall effects and also high-light existing degeneracies between changes caused by αEM, me andthe average CMB temperature, T0. In Sect. 4, we present our con-straints for different cases using Planck 2015 data. In particular,we find the CMB data to be sensitive not only to the value of thefundamental constants around recombination but also its first timederivative. We conclude in Sect. 5.

2 EFFECTS OF VARYING FUNDAMENTALCONSTANTS ON IONIZATION HISTORY

In this section, we describe the effects of varying αEM and me onthe ionization history. We use modified versions of CosmoRec andRecfast++ (Chluba et al. 2010) for our computations1, highlight-ing the importance of different effects and their individual impacton the free electron fraction, Xe.

2.1 How do αEM and me enter the recombination problem?

Varying αEM and me inevitably creates changes in the ionizationhistory. Most importantly, the energy levels of hydrogen and heliumdepend on these constants, Ei ∝ α

2EMme, which directly affects the

recombination redshift. In addition, atomic transition rates and pho-toionization/recombination rates are altered. Lastly, the interactionsof photons and electrons through Compton and resonance scatter-ing modify the radiative transfer physics. In an effective three-levelatom approach (Zeldovich et al. 1968; Peebles 1968; Seager et al.2000), the individual dependencies can be summarized as (e.g.,Kaplinghat et al. 1999; Scoccola et al. 2009; Planck Collaborationet al. 2015b; Chluba & Ali-Haımoud 2016)

σT ∝ α2EMm−2

e A2γ ∝ α8EMme PSA1γ ∝ α

6EMm3

e

αrec ∝ α2EMm−2

e βphot ∝ α5EMme Teff ∝ α

−2EMm−1

e .(1)

Here, σT denotes the Thomson scattering cross section; A2γ is thetwo-photon decay rate of the second shell; αrec and βphot are the ef-fective recombination and photoionization rates, respectively; Teff

is the effective temperature at which αrec and βphot need to be eval-uated (see explanation below); PSA1γ denotes the effective dipoletransition rate for the main resonances (e.g., Lyman-α), which is

1 These codes are available at www.Chluba.de/CosmoRec.

500 1000 1500 2000 2500 3000z

-4

-3

-2

-1

0

1

2

∆Xe/X

e in

%

TotalHHe

Figure 1. Relative difference in the free electron fractions of CosmoRec andRecfast++ (Recfast++ is used as reference) for the standard cosmology.The lines show the results for Xe = XH

e + XHee (black/solid), XH

e (dotted/red)and XHe

e (dashed/blue).

reduced by the Sobolev escape probability, PS ≤ 1 (Sobolev 1960;Seager et al. 2000) with respect to the vacuum rate, A1γ. For a moredetailed account of how the transition rates depend on the funda-mental constants we refer to Chluba & Ali-Haımoud (2016) andthe manual of HyRec (Ali-Haımoud & Hirata 2011).

The scalings of σT, A2γ and PSA1γ directly follow from theirexplicit dependencies on αEM and me. The shown scalings of αrec

and βphot reflect renormalisations of the transition rates, again stem-ming from their explicit dependencies on αEM and me (e.g., Karzas& Latter 1961). However, these rates also depend directly on theratio of the electron/photon temperature to the ionization thresh-old. This leads to an additional dependence on αEM and me, whichcan be captured by evaluating these rates at rescaled temperature,with scaling indicated through Teff . Overall, this leads to the effec-tive dependence αrec ∝ α

3.44EM m−1.28

e around hydrogen recombination(Chluba & Ali-Haımoud 2016). The required photoionization rate,βphot, is obtained using the detailed balance relation. Slightly dif-ferent overall scalings for αrec and βphot were used in Planck Col-laboration et al. (2015b), but we find the associated effect on therecombination history to be sub-dominant and limited to z . 800.

We also highlight that all atomic species are treated using hy-drogenic scalings. For neutral helium, non-hydrogenic effects (e.g.,fine-structure transitions, singlet-triplet couplings) become relevant(Drake 2006). However, the corrections should be sub-dominantand are neglected here.

We will illustrate the importance of the different terms inEq. (1) in Sect. 2.2. This will show that in particular the changes inthe energy scale, which are captured by rescaling the temperature,are crucial. We now continue by explaining the required modifica-tions to Recfast++ and CosmoRec.

2.1.1 Modifications to Recfast++

Recfast++ is based on a simple three-level atom approach, similarto that of recfast (Seager et al. 1999). It evaluates three ordinarydifferential equations, evolving the free electron fraction contribu-tions from hydrogen and singly-ionized helium, XH

e and XHee , re-

spectively, as well as the matter/electron temperature, Te. Doubly-ionized helium is modeled using the Saha-relations.

As an added feature of the Recfast++ code, one can mod-

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Variation of αEM and me 3

ify the obtained ionization history with a correction function torepresent the full recombination calculation of CosmoRec (Rubino-Martın et al. 2010b; Shaw & Chluba 2011). The required correctionfunction between Recfast++ and CosmoRec is obtained as

XCe (z) ≈

(1 +

∆Xe(z)XR

e (z)

)XR

e (z) = ftot(z) XRe (z), (2)

where ’C’ refers to CosmoRec, ’R’ to Recfast++. In the code, therelative difference, ∆Xe/XR

e = (XCe − XR

e )/XRe , is stored for the stan-

dard cosmology and then interpolated to obtain ftot(z). The relativedifference, ∆Xe/XR

e , is illustrated in Fig. 1. For the standard cos-mology, Recfast++ naturally allows a quasi-exact representationof the full calculation. For small variations around the standard cos-mology, the correction-to-correction can be neglected so that thisapproach remains accurate in CMB analysis (Shaw & Chluba 2011;Planck Collaboration et al. 2015a).

All the modifications listed in Eq. (1) are readily incorporatedto the simple Recfast++ equations. However, we found that forour purpose it was beneficial to treat the correction functions forhydrogen and helium separately, since in the transition regime be-tween hydrogen and helium recombination (z ' 1600) the free elec-tron fraction departs from unity, which is physically not expected.This generalizes Eq. (2) to

XCe (z) ≈ fH(z) XH,R

e (z) + fHe(z) XHe,Re (z), (3)

where we multiply each correction function term with its respectivecontribution to the total Xe = XH

e + XHee . The individual correction

functions are again obtained using relative differences with respectto the standard cosmology, fi(z) = 1 + ∆Xi

e/Xi,Re . This is illustrated

in Figure 1. At z ' 2500, the helium correction sharply drops to∆XHe

e /XHe,Re ≈ −100% (i.e. fHe → 0), indicating that helium rapidly

recombines. This is related to hydrogen continuum absorption ofhelium photons, which is not taken into account in the standardtreatment (Kholupenko et al. 2007; Switzer & Hirata 2008; Rubino-Martın et al. 2008). Since hydrogen recombination occurs at lowerredshifts, the hydrogen corrections tend to 0 at z & 1500, whilearound z ' 1100 radiative transfer corrections become visible (e.g.,Fendt et al. 2009; Rubino-Martın et al. 2010b, for overview).

At z . 1500, ftot(z) ≈ fH(z), while the features related to he-lium recombination corrections around z ' 1700 are now repre-sented directly by the helium correction function. Once added toRecfast++, it more fairly weights the helium corrections than theprevious approach. In the code, one can chose between the two ver-sions, but we find that when varying the fundamental constants, thenew approach works best. It is furthermore important to interpretthe correction functions as a function of temperature. This leads tothe remapping z → z × (αEM/αEM,0)−2(me/me,0)−1, which capturesthe leading order transformation of radiative transfer corrections.

2.1.2 Modifications to CosmoRec

The modifications to Recfast++ were relatively straightforward.However, for CosmoRec this became a slightly bigger task.CosmoRec is built up as a modular system that allows each mod-ule to act as a plugin. In CosmoRec, the energies and transitionrates within the hydrogen and neutral helium atoms needed to berescaled with the previously mentioned scalings. These are repre-sented by classes called Atom and HeI Atom, respectively, whichinclude all the properties of given atomic levels, the collection oflevels that form the atom and the ensemble of atoms around recom-bination. These can also be used as independent coding modulesfor atomic physics calculations. The neutral helium scalings with

0 1000 2000 3000 4000 5000 6000 7000 8000z

0

0.2

0.4

0.6

0.8

1

1.2

X e

∆α/α = -0.1∆α/α = -0.05∆α/α = 0∆α/α = 0.05∆α/α = 0.1

Figure 2. Ionization histories for different values of αEM. The dominant ef-fect is caused by modifications of the ionization threshold, which impliesthat for increased αEM recombination finishes earlier. The curves were com-puted using Recfast++.

αEM and me are modeled using hydrogenic expressions, which isexpected to be accurate at the ' 0.1% − 1% level but omits higherorder effects to the energy levels or transition rates.

After the atomic initializations, the effective transition rates(see Ali-Haımoud & Hirata 2010, for details about the method) re-lated to the multi-level atom need to be rescaled. In the code, theseaffect the effective recombination rates,A(Tγ,Te), the photoioniza-tion rates, B(Tγ) and the inter-state transition rates, R(Tγ). Changesrelated to σT are again trivial to include.

During recombination, the processes occurring within theatoms are influenced by the temporal evolution of the backgroundphoton field. This complicates the recombination problem with theneed for partial differential equations (PDEs) describing the radia-tive transfer (e.g. Chluba & Sunyaev 2007; Grachev & Dubrovich2008; Chluba & Sunyaev 2009b; Hirata 2008; Hirata & Forbes2009; Chluba & Sunyaev 2009a). When the fundamental constantsare modified, one must again rescale the rates and energies requiredfor the computations of the photon field. Similarly, the two-photonand Raman scattering profiles (Chluba & Sunyaev 2008; Hirata2008; Chluba & Thomas 2011) have to be altered. We also care-fully considered modifications to the neutral helium radiative trans-fer (Chluba et al. 2012). These effects can be separately activatedin the latest version of CosmoRec (i.e. version 3.0 or higher).

2.2 Relevance of different effects for Xe

We now illustrate the importance of the individual effects in Eq. (1),for now assuming constant changes of αEM and me. This will begeneralized in Sect. 2.3. We shall start by focusing on changescaused by varying αEM, parametrized as αEM = αEM,0(1 + ∆α/α).When all the terms relevant to the recombination problem are in-cluded, we obtain the ionization histories shown in Fig. 2 for differ-ent values of ∆α/α. Increasing the fine structure constant shifts themoment of recombination toward higher redshifts. This agrees withthe results found earlier in Kaplinghat et al. (1999), Battye et al.(2001) and Rocha et al. (2004) and can intuitively be understood inthe following manner: ∆α/α > 0 increases the transition energiesbetween different atomic levels and the continuum. This increasesthe energy threshold at which recombination occurs, hence increas-

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4 L. Hart and J. Chluba

0 500 1000 1500 2000 2500 3000z

-3

-2.5

-2

-1.5

-1

-0.5

0

∆Xe/X

e in

%

σT x 10A2γ

αrec and βphotPesc A1γ

T (Boltzmann)all

∆α/α = 10-3

Figure 3. The relative changes in the ionization history for ∆α/α = 10−3

with respect to the standard case caused by different effects. Recfast++was used for the computations. The rescaling of temperature (↔ mainlyaffecting the Boltzmann factors) yields ∆Xe/Xe ' −2.7%, dominating thetotal contributions, which peaks with ' −3.1% at z ' 1000. Note that themodification due to σT has been scaled by 10 to make it visible.

ing the recombination redshift, an effect that is basically capturedby an effective temperature rescaling in the evaluation of the pho-toionization and recombination rates (see below).

The relative changes to the ionization history, ∆Xe/Xe, for thedifferent terms discussed in Section 2.1 are illustrated in Fig. 3. Wechose a value for ∆α/α = 10−3, which leads to a percent-level effecton Xe. As expected, the biggest effect appears after rescaling thetemperature for the evaluation of the photonionization and recom-bination rates. More explicitly, this can be understood when consid-ering the net recombination rate to the second shell, which can bewritten as ∆Rcon = NeNpαrec − N2 βphot = αrec[NeNp − g(Tγ)N2](in full equilibrium, ∆Rcon = 0), where g(Tγ) ∝ T 3/2

γ e−hν2c/kTγ

with continuum threshold energy, E2c = hν2c. Here, the expo-nential factor (↔ Boltzmann factor) is most important, leadingto an enhanced effect once the replacement T ′γ(z) = Tγ(z) ×(αEM/αEM,0)−2(me/me,0)−1 is carried out. For ∆α/α = 10−3, thisgives ∆Xe/Xe ' −2.7% at z ' 1000, which accounts for nearlyall of the effect (cf., Fig. 3).

The second largest individual effect is due to the rescaling ofthe two-photon decay rate, A2γ. This is expected since αEM appearsin a high power, A2γ ∝ α8

EM, and because the 2s-1s two-photonchannel plays such a crucial role for the recombination dynamics(Zeldovich et al. 1968; Peebles 1968; Chluba & Sunyaev 2006b),allowing ' 58% of all hydrogen atoms to become neutral throughthis route (Chluba & Sunyaev 2006a). For ∆α/α = 10−3, we find∆Xe/Xe ' −0.5% at z ' 1000.

The normalizations of the recombination and photoionizationrates (blue/dashed line) give rise to a net delay of ∆Xe/Xe ' 0.3%at z ' 1000, which partially cancels the correction due to A2γ. Thisis due to the stronger scaling of βphot with αEM than αrec. At low red-shifts (z . 750), recombination is again accelerated, indicating thata higher fraction of recombination events occurs, as the importanceof photoionization ceases. The correction related to the Lyman-αchannel is found to be ' 3.3 times smaller than for the two-photonchannel, yielding ∆Xe/Xe ' −0.15% at z ' 1000 (cf., Fig. 3).

Figure 3 also shows that the contributions from rescaling σT

are very small and only become noticeable at low redshifts. At theseredshifts, the matter and radiation temperature begins to depart, giv-

0 500 1000 1500 2000 2500 3000z

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

∆Xe/X

e in

%

σT x 10A2γ

αrec and βphotPesc A1γ

T (Boltzmann)all

∆me/me = 10-3

Figure 4. Same as in Fig. 3 but for ∆me/me = 10−3. The effective tempera-ture rescaling again dominates the total change. Around z ' 1000, the totaleffect is ' 2.5 times smaller than for ∆α/α = 10−3.

0 500 1000 1500 2000 2500 3000z

-3

-2.5

-2

-1.5

-1

-0.5

0

∆Xe/X

e in

%

∆α/α = 10-3 in Recfast++∆α/α = 10-3 in CosmoRec∆me/me = 10-3 in Recfast++

∆me/me = 10-3 in CosmoRec

difference in the freeze out tail

x 2.5

Figure 5. Comparison of the changes to the ionization history caused byvariation of αEM and me as computed with Recfast++ and CosmoRec. Bothcodes agree extremely well (the lines practically overlap for the two cases),departing only at the level of ' 0.1% for Xe. For changes of me, the effecton the freeze-out tail is much smaller than for αEM.

ing Te < Tγ. For larger αEM, this departure is delayed, such that Te

stays longer close to Tγ. Hotter electrons recombine less efficiently,so that a slight delay of recombination appears (cf., Fig. 3). We findthat this correction can in principle be neglected without affectingthe results notably, but include it for completeness.

2.2.1 Changes due to variation of me

We now focus on changes caused by the effective electron mass,parametrized as me = me,0(1 + ∆me/me). Inspecting the scalings ofEq. (1), we expect the overall effect to be smaller than for αEM. Forexample, the effect of temperature rescaling should be roughly halfas large. Similarly, the effect due to rescaling A2γ should be roughly8 times smaller, and so on. This is in good agreement with our find-ings (cf. Fig. 4). The net effect on Xe is about 2.5 times smaller thanfor αEM around z ' 1000 (see Fig. 5 for a direct comparison). Thissuggests that the CMB constraint on me is weakened by a similarfactor. However, adding the rescaling of the Thomson cross section

c© 2017 RAS, MNRAS 000, 1–12

Variation of αEM and me 5

0 2000 4000 6000 8000 10000z

0

0.2

0.4

0.6

0.8

1

1.2X e

700 800 900 1000 1100 120010-3

10-2

10-1

100

p = -0.1p = 0p = 0.1

Figure 6. Ionization histories for redshift-dependent variation of thefine-structure constant, αEM(z) = αEM(z0) [(1 + z)/1100]p. Here, we setαEM(z0) to the standard value, αEM(z0) ≈ 1/137, and only varied p. Thedifferent phases in the ionization history are stretched/compressed with re-spect to the standard case, depending on the chosen value for p , 0.

for the computation of the visibility function strongly enhances ge-ometric degeneracies for me, such that the CMB only constraint onme is & 20 times weaker than for αEM (see Sect. 3.1.3).

A small difference related to the renormalizations of the pho-toionization and recombination rates (blue/dashed line) appears.For ∆me/me > 0, the photoionization rate is increased and the re-combination rate is reduced for these contributions [cf. Eq. (1)].Both effects delay recombination (see Fig. 4). Thus, around z ' 103

the net effect is slightly larger than for αEM. In contrast to αEM, atlate time no net acceleration of recombination occurs. These ef-fects slightly modify the overall redshift dependence of the totalXe change, in addition lowering the effect in the freeze-out tail (seeFig. 5 for direct comparison). At the level ∆me/me ' 1%, additionalhigher order terms become important, allowing one to break thedegeneracy between αEM and me in joint analyses (see also PlanckCollaboration et al. 2015b).

We note that we ignored the extra ρb/me scaling in the Comp-ton cooling term for Te, rescaling me in the atomic quantitiesonly. When varying fundamental constants, dimensionless vari-ables should furthermore be used (e.g., ?), so that an analysis ofexplicit me variations remains phenomenological.

2.2.2 Comparing Recfast++ and CosmoRec

We close by directly comparing the results for Xe obtained withRecfast++ and CosmoRec (Fig. 5). Both codes agree extremelywell, departing by . 0.1% in Xe. Tiny differences in the resultant∆Xe/Xe are visible around helium recombination (z ' 1700), whichare related to radiative transfer effects that CosmoRec models ex-plicitly. Similarly, around the maximum of the Thomson visibil-ity function (z ' 1100), small percent-level differences in ∆Xe/Xe

are present. These differences do not affect the computation of theCMB anisotropies at a significant level and thus our Recfast++treatment is sufficient for the analysis presented in Sect. 4. We ex-plicitly confirmed this by comparing the constraints obtained withthe two recombination codes for αEM and me, finding them to agreeto high precision. Similarly, for the analysis of future CMB data(e.g. CMB Stage-IV), we deem our treatment with Recfast++ tosuffice in these cases.

600 800 1000 1200 1400z

0

1

2

3

4

5

g(z)

x 1

03

∆α/α = -0.05∆α/α = 0∆α/α = 0.05

Figure 7. Visibility functions for a variety of fine structure constant values.A higher value of the fine structure constant leads to a broader visibilityfunction, which simultaneously reduces its height. For illustration, the dot-ted lines exclude the rescaling of σT within CAMB.

2.3 Adding an explicit redshift dependence to the variations

We extend our treatment of variation of fundamental constants byalso considering an explicit redshift-dependence of αEM and me,assuming a phenomenological power-law scaling around pivot red-shift2 z0 = 1100. This could in principle be caused by the presenceof a scalar field and its coupling to the standard particle sector dur-ing recombination. For αEM, our model reads

αEM(z) = αEM(z0)(

1 + z1100

)p

, (4)

and similarly for me. For p 1, we find a logarithmic de-pendence on redshift, αEM(z) ≈ αEM(z0) (1 + p ln[(1 + z)/1100]).Note that the rescaled value at the pivot redshift is not necessarilyαEM(z0) ≡ αEM,0 ' 1/137, but can also be varied. Here, p is a vari-able index that determines how the ionization history is stretched orcompressed around the central redshift. We added this new optionto Recfast++. Some examples are shown in Fig. 6. For p > 0, re-combination is accelerated at z & 1000 with respect to the standardcase, while it is delayed at z . 1000. For p , 0, due to cumula-tive effects the change in Xe does not vanish at the pivot redshift.Also, the modification is very different to that of a constant shift ofαEM, predominantly affecting the width of the Thomson visibilityfunction as opposed to the position (see Sect. 3). Thus, geometricdegeneracies are found to be less important when constraining thevalue of p using CMB data (Sect. 4).

3 PROPAGATING THE EFFECTS TO THE CMBANISOTROPIES

The temperature and polarization power spectra of the CMB de-pend on the dynamics of recombination through the ionization his-tory, which defines the Thomson visibility function and last scat-tering surface (e.g., Sunyaev & Zeldovich 1970; Peebles & Yu1970; Hu & Sugiyama 1996). Therefore, when varying αEM andme, this leads to changes in the CMB power spectra. In this section,we show the modifications of the Thomson visibility function forthe effects discussed in Section 2. The modified CMB temperature

2 This choice de-correlates redshift-dependent and constant changes.

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6 L. Hart and J. Chluba

power spectra are then computed using CAMB (Lewis et al. 2000)for the standard cosmology (Planck Collaboration et al. 2015a).We briefly explain the main effects on the power spectra due tovarying fundamental constants. Excluding modifications in the re-combination dynamics, the CMB anisotropies still directly dependon the Thomson scattering cross section. We show that the changesfrom rescaling σT explicitly within CAMB are much smaller thanthose caused by modifications to the recombination dynamics. Still,they need to be included when deriving CMB constraints on fun-damental constants (see also Planck Collaboration et al. 2015b), inparticular when studying variations of me (see Sect. 4). We alsopresent the changes of the CMB temperature power spectrum forthe redshift-dependent variations of αEM and me from Section 2.3.

3.1 Changes due to constant shifts of αEM and me

Using the result for the ionization history computed with the modi-fied version of Recfast++, one can calculate the Thomson visibil-ity function, g(z), defined as,

g(z) =dτdz

exp [−τ(z)] . (5)

Here, dτ/ dz is the differential Thomson optical depth. The Thom-son visibility function can be interpreted as an effective probabil-ity distribution for a photon being last-scattered around redshift z.It is normalized such that

∫g(z) dz = 1. From the changes in Xe

described above, we expect that for ∆α/α > 0 the maximum ofthe visibility function shifts toward higher redshifts. In Fig. 7, thevisibility function is shown for constant ∆α/α = −0.05, 0, 0.05.Indeed, the visibility function maximum moves to zmax ≈ 1200 for∆α/α = 0.05. The relative width, ∆z FWHM/z max, of the visibilityfunction is roughly conserved.

3.1.1 Effects on the CMB anisotropies due to variations of αEM

We illustrate the modifications to the CMB power spectrum forconstant changes of αEM in Fig. 8. We focus on the CMB tempera-ture power spectra, as the effects on the polarization power spectraare qualitatively similar. Two main effects are visible. Firstly, thepeaks of the power spectrum are shifted to smaller scales (larger`) when ∆α/α > 0. This happens because earlier recombinationmoves the last scattering surface towards higher redshifts, whichdecreases the sound horizon and increases the angular diameter dis-tance to recombination (Kaplinghat et al. 1999; Battye et al. 2001).Secondly, for ∆α/α > 0, the peak amplitudes are enhanced. This ismainly because earlier recombination suppresses the effect of pho-ton diffusion damping on the anisotropies (Kaplinghat et al. 1999;Battye et al. 2001). For small ∆α/α, we also illustrate the relativechange of the temperature power spectrum in Fig. 10. The effecton the peak positions is more noticeable than the small overall tiltcaused by changes related to diffusion damping.

3.1.2 Separate effect related to σT

The Thomson scattering cross section, σT, enters the problem intwo ways. Firstly, it directly affects the recombination dynamicsand thermal coupling between photons and baryons, as explainedabove (Sect. 2.2). These changes are taken into account when com-puting the recombination history, but turn out to be minor (Fig. 3and 4) and can in principle be neglected. Secondly, σT also directlyappears in the definition of the Thomson visibility function, g(z),which is computed inside CAMB and has to be modified separately(see Kaplinghat et al. 1999; Planck Collaboration et al. 2015b).The comparably small effect on g(z) is illustrated in Fig. 7 for

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∆α/α = ±0.05, where the dotted lines exclude the rescaling of σT

within CAMB. The corresponding changes to the TT power spec-trum for ∆α/α = 0.05 are shown in Fig. 9. In the considered `

range, the maximal relative difference is |∆C`/C` | ' 6%, which ismore than one order of magnitude smaller than the effects due todirect changes in Xe discussed above. However, in particular whenstudying variations of me, this effect has to be included as otherwisethe errors are strongly underestimated (see Sect. 4).

We included the effect of σT rescaling for the computation ofthe visibility in two independent ways. First, we consistently imple-mented these changes into CAMB by adding a rescaling function thattargets the akthom components and Compton cooling terms withinmodules.f90 and reionization.f90. Second, we simply rede-fined the free electron fraction, Xe returned by the recombinationcode to CAMB as X∗e = (αEM/αEM,0)2(me/me,0)−2 Xe. The two ap-proaches gave extremely similar results for the power spectra andalso final parameter constraints. The only real difference is that inthe first approach, the corrections to the reionization history are in-cluded more consistently, albeit not being modeled in a physicalmanner. For example, for ∆α/α > 0, the reionization redshift re-

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Figure 10. Comparison of the CMB TT power spectrum deviations whenvarying αEM, me, T0 and p. We chose ∆α/α = 10−3, ∆me/me = 2 × 10−3,∆T/T = −10−2 and p = 5×10−3 (simultaneously for αEM and me) to obtaineffects at a similar level. Notice the small extra tilt when comparing the casefor αEM with me, which helps when constraining αEM.

duces for fixed value of τ. In the second approach, the reionizationhistory is not affected but the correction is minor. For our analysis,we used the explicit rescaling in CAMB including all terms.

3.1.3 Effects on the CMB anisotropies due to variations of me

We now briefly mention the changes caused by variation of me.As discussed above, for the free electron fraction the net changesare very similar to those for αEM. Thus, one expects both changesin the visibility and CMB power spectra to be similar, albeit ata lower amplitude when ∆me/me ' ∆α/α 1. Indeed, wefind the changes in the visibility function around its maximumto mimic those shown in Fig. 7 for variations of αEM when set-ting ∆me/me ' (2 − 3) × ∆α/α. This is expected when compar-ing the main effect on Xe around redshift z ' 103 for αEM andme (Fig. 3 and Fig. 4), and suggests that the me-related changesin the CMB power spectra are also weakened by a similar fac-tor. This is explicitly illustrated in Fig. 10, which shows that asidefrom a small overall tilt the changes in the CMB TT power spec-tra, ∆C`/C`(∆α/α) and ∆C`/C`(∆me/me), become almost indistin-guishable when using ∆me/me ≈ (2 − 3) ∆α/α. This presents aquasi-degeneracy between the two parameters and also suggeststhat naively the analysis for ∆α/α could be sufficient to estimatethe errors for a corresponding analysis of me. However, when con-straining me, enhanced geometric degeneracies (because of σT)push the error to the percent level. In this case, higher order termsbecome important and the degeneracy is again broken. When alsoadding information from BAO, the error on me is strongly reduced.In this regime, we indeed recover the simple scaling of the errors,σ(∆me/me) ' 3σ(∆α/α) [see Table 2].

3.1.4 Degeneracies between αEM and T0

Our previous discussion showed that a variation of αEM and me di-rectly affect the recombination redshift. The main effect can be cap-tured by rescaling the Boltzmann factors using Teff . This suggeststhat a change in the CMB monopole temperature, T0, could have avery similar effect. However, there is one crucial difference: T0 alsoaffects the matter-radiation equality epoch. This modifies the inte-grated Sachs Wolfe (ISW) effect, which is most noticeable at lowand intermediate ` and in principle should help one to break the

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Figure 12. The CMB power spectrum for various values of the redshiftpower law index p for αEM. Positive values of the p index lead to a sup-pression of the CMB peaks due to broadening of the recombination epoch.

degeneracy between changes of αEM and T0. The relative changeto the TT power spectrum is illustrated in Fig. 10, where we use∆T0/T0 ' −0.01 to make the modifications comparable in ampli-tude. One can clearly see the enhanced effect at large angular scalesdue to the early and late ISW (see Fig. 3 in ?, for illustration of thesecontributions).

3.2 Changes due to redshift-dependent variations

We now consider redshift-dependent variations to αEM and me, us-ing the parametrization given by Eq. (4). We assume the stan-dard values for αEM and me at z0 = 1100. In Fig. 11, we illus-trate the effect on the Thomson visibility function for αEM. Usingp < 0, the visibility function narrows such that the effective width,∆z FWHM/z max, reduces. However, the changes in the position ofthe maximum value of g(z) are negligible. This suggests that thechanges in the positions of the peaks in the CMB power spectra areminor, while the blurring related to the finite thickness of the last

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8 L. Hart and J. Chluba

Parameter Planck 2015 + varying αEM/αEM,0 + varying p + varying αEM/αEM,0 and p

Ωbh2 0.02224 ± 0.00016 0.02225 ± 0.00016 0.02226 ± 0.00018 0.02223 ± 0.00019Ωch2 0.1193 ± 0.0014 0.1191 ± 0.0018 0.1194 ± 0.0014 0.1193 ± 0.0020100θMC 1.0408 ± 0.0003 1.0398 ± 0.0035 1.0408 ± 0.0003 1.0406 ± 0.0051τ 0.062 ± 0.014 0.063 ± 0.014 0.062 ± 0.014 0.063 ± 0.015ln(1010As) 3.057 ± 0.025 3.060 ± 0.027 3.058 ± 0.026 3.059 ± 0.027ns 0.9649 ± 0.0047 0.9668 ± 0.0081 0.9663 ± 0.0060 0.9666 ± 0.0081

αEM/αEM,0 – 0.9993 ± 0.0025 – 0.9998 ± 0.0036p – – 0.0008 ± 0.0025 0.0007 ± 0.0036

H0 [km s−1 Mpc−1] 67.5 ± 0.6 67.2 ± 1.0 67.5 ± 0.6 67.3 ± 1.4

Table 1. Constraints on the standard ΛCDM parameters and the fundamental constant parameters αEM/αEM,0 and p for different combinations of parameters.The standard Planck runs include the TTT EEE likelihood along with the low ` polarization and CMB lensing likelihoods and the errors are the 68% limits.

scattering surface is reduced3. The separate effect of rescaling σT

inside CAMB is also illustrated in Fig. 11, an effect that we find tohave a negligible impact when constraining the value of p alone.Similar comments apply for changes to me.

We show the changes in the CMB temperature power spec-tra due to redshift-dependent variations of αEM in Fig. 12. Whenwe choose p < 0, the CMB peaks are amplified. This is expectedfrom the reduced width, ∆z FWHM/z max, of the visibility function inFig. 11. Similarly, for p > 0, a larger damping effect due to blurringis found. The relative change of C` for p = 5 × 10−3 is shown inFig. 10. The smoothness of the titled curve indicates that blurringof anisotropies is indeed the dominant effect. Again, similar effectsare found for changes to me.

4 CONSTRAINTS USING PLANCK DATA

We now constrain the variations of αEM, me and p discussed inSec. 3 using CosmoMC (Lewis & Bridle 2002) with the Planck 2015data4. We sample over the acoustic angular scale, θMC. Although forthis specific analysis, H0 is expected to de-correlate quicker, we didnot encounter any problems. We find that the constraints derived forαEM and me are consistent with those for Planck 2013 data (PlanckCollaboration et al. 2015b), albeit here with slightly improved er-rors. We also show the new constraints for our redshift dependentmodel of αEM. Our marginalized constraints are summarized in Ta-bles 1 and 2. For comparison, the standard 6 ΛCDM parameter runfor the Planck data is also given. For each run, we show the derivedH0 parameter. The 2D parameter contours are shown in Fig. 13.

4.1 Constraining αEM and me

When varying αEM, assuming constant ∆α/α, along with the 6 stan-dard cosmological parameters, we find the marginalized parametervalues in the second column of Table 1. These show that αEM/αEM,0

is equal to unity well within the 68% limit. The errors on θMC in-creases by about one order of magnitude, due to the added uncer-tainty in the distance to the last scattering surface. We also find aslight increase in the errors of the scalar spectral index, ns, whichinteracts with the modifications to the photon diffusion dampingscale caused by αEM. Similarly, the error of the cold dark matterdensity, Ωch2, increases slightly, due to geometric degeneracies.

3 An explanation of this damping effect can be found in Mukhanov (2004).4 When quoting Planck 2015 data we usually refer to the likeli-hood Planck 2015 TTT EEE+lowP+lensing. For Planck 2013, we implyPlanck+WP+lensing as baseline.

The other parameters (i.e., τ and As) are largely unaltered by theaddition of αEM as a parameter (see Fig. 13). This highlights thestability and consistency of the data with respect to non-standardextensions of the cosmological model.

Although the contributions from σT appear to have a negli-gible effect on the C` (see Fig. 9), we find that the inclusion ofthis effect improves the errors on αEM by ' 30%. The small effecton the power spectra hinders some of the degeneracy between θMC

and αEM/αEM,0, as pointed out in previous analyses (e.g. PlanckCollaboration et al. 2015b). For example, the marginalized value ofαEM/αEM,0 changes from 0.9988± 0.0033 to 0.9993± 0.0025 whenincluding σT rescaling within CAMB. The former result is consistentwith the one presented recently in Di Valentino et al. (2016), whichsuggests that in their analysis this modification was neglected.

For constant changes to me, we obtain the results given in thefirst two columns of Table 2. Using 2015 CMB data alone, wefind me/me,0 = 0.961+0.046

−0.072 and H0 = 60+ 8−16 km s−1 Mpc−1, which

is consistent with the corresponding result (Planck+WP+lensing),me/me,0 = 0.969 ± 0.055 and H0 = (62 ± 10) km s−1 Mpc−1, givenin Planck Collaboration et al. (2015b). Following the Planck 2013analysis, we used a flat prior H0 = [40, 100] km s−1 Mpc−1. We notethat at the lower end of this range this leads to a slight trunca-tion of the posterior distribution for H0. Also, our errors for theCMB-only analysis remain asymmetric, even when repeating thePlanck 2013 run, for which we find me/me,0 = 0.964+0.054

−0.068 andH0 = 61+9.5

−15 km s−1 Mpc−1. We confirmed that the remaining dif-ference is not related to the slightly different scalings for αrec andβphot used in Planck Collaboration et al. (2015b).

For varying me, the values of H0 and me are both biasedlow when only using CMB data (see also Planck Collaborationet al. 2015b). Interestingly, this bias is removed when neglect-ing the effect of σT on the Thomson visibility, for which we findH0 = (67.0 ± 1.6) km s−1 Mpc−1 and me/me,0 = 0.9970 ± 0.0098.This treatment also significantly decreases the errors due to reducedgeometric degeneracies, which highlights the importance of σT forthe computation of constraints on variations of me. At the level of∆me/me ' 1%, also non-linear corrections become noticeable.

Given the large geometric degeneracy, we also ran the con-straint for me when adding BAO data (see Table 2). In this case,we obtained5 me/me,0 = 1.0039 ± 0.0074, which, albeit improvederror, is consistent with the result me/me,0 = 1.004± 0.011 given inPlanck Collaboration et al. (2015b). The value of H0 returns to thestandard CMB value when adding BAO data. For this combination

5 Adding lensing did not affect the constraint at a very significant level.

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Figure 13. The full contours with the TTT EEE + lowP + lensing likelihoods from Planck 2015. The standard ΛCDM run (black, dotted) is shown alongsidethe added variations in αEM (blue), p (red) and the combination of the two parameters (green).

Parameter Planck 2015 Planck 2015 + BAO Planck 2015 Planck 2015 + BAO+ me/me,0 + me/me,0 + αEM/αEM,0 and me/me,0 + αEM/αEM,0 and me/me,0

Ωbh2 0.0213+ 0.0011− 0.0017 0.02233 ± 0.00018 0.0214+ 0.00099

− 0.0017 0.02238 ± 0.00020Ωch2 0.1146+ 0.0059

− 0.0087 0.1202 ± 0.0022 0.1144+ 0.0057− 0.0090 0.1200 ± 0.0023

θMC 1.012+ 0.036− 0.051 1.0435 ± 0.0052 1.011+ 0.034

− 0.054 1.0431 ± 0.0053τ 0.057 ± 0.015 0.080 ± 0.017 0.058 ± 0.015 0.082 ± 0.019ln(1010As) 3.044 ± 0.029 3.095 ± 0.033 3.048 ± 0.031 3.100 ± 0.037ns 0.9639 ± 0.0048 0.9647 ± 0.0046 0.9663 ± 0.0078 0.9678 ± 0.0085

αEM/αEM, 0 – – 0.9990 ± 0.0025 0.9989 ± 0.0026me/me,0 0.961+ 0.046

− 0.072 1.0039 ± 0.0074 0.962+ 0.044− 0.074 1.0056 ± 0.0080

H0 [km s−1 Mpc−1] 60+ 8− 16 68.1 ± 1.3 60+ 7

− 16 68.1 ± 1.3

Table 2. Constraints on the standard ΛCDM parameters and the fundamental constant parameters αEM/αEM,0 and me/me,0 for different combinations ofparameters. The standard Planck runs include the TTT EEE likelihood along with the low ` polarization and CMB lensing likelihoods and the errors are the68% limits. We also show the results when adding BAO data.

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10 L. Hart and J. Chluba

0.984 0.988 0.992 0.996 1.000 1.004EM/ EM, 0

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0Planck 2013Planck 2015

Planck 2015 + BAO

Figure 14. The joint likelihood contour for me and αEM for Planck2013+WP+lensing and Planck 2015 TTT EEE+lowP+lensing data. ThePlanck 2015 contours are narrowed in the αEM direction due to improvedpolarization information over Planck 2013. Adding BAO data to Planck2015 further improves the constraint in particular on me. The dashed linesindicate ∆me/me = ∆α/α = 0 for reference.

of datasets, we also find that the error on me is ' 3 times largerthan the corresponding error on αEM, as naively expected from thesimilarity of the changes in the TT power spectrum (Fig. 10).

4.1.1 Simultaneously constraining αEM and me

We finish our analysis of this section by simultaneously varyingαEM and me (see Table 2 for our constraints). The responses in theCMB power spectra are quite similar for ∆α/α ' ∆me/me ' 10−3

(see Fig. 10), suggesting a significant degeneracy between αEM andme. However, combined CMB-only constraints are obtained whennon-linear corrections in particular for me become noticeable, sothat both parameters can be simultaneously constrained.

The strong degeneracies between αEM and me are substan-tially reduced when going from WMAP to Planck 2013, as al-ready described in Planck Collaboration et al. (2015b). In Fig. 14,we show our contours for Planck 2013 and 2015 data. The non-Gaussian shapes of the contours are reminiscent of the non-linearterms mentioned above. We find αEM/αEM,0 = 0.9990 ± 0.0025and me/me,0 = 0.962+0.044

−0.074 for Planck 2015. This improves overour constraint for Planck 2013, αEM/αEM,0 = 0.9936 ± 0.0042 andme/me,0 = 0.977+0.056

−0.071, which is in good agreement with the resultgiven in Planck Collaboration et al. (2015b) for this case. The im-provement is mainly due to better polarization information.

As for the analysis of me, we can see that CMB data alonetends towards low values of H0 and me/me,0. This bias is removedwhen adding BAO, for which we find αEM/αEM,0 = 0.9989±0.0026,me/me,0 = 1.0056±0.0080 and H0 = 68.1±1.3. These numbers areconsistent with the standard Planck 2015 cosmology (see Table 2).

4.2 Constraining the redshift dependence of αEM and me

Next, we consider redshift-dependent variations of αEM, using theparametrization in Eq. (4) with αEM(z0) = αEM,0. The constraintsfor this case are shown in third column of Table 1. Since varying pmainly affects the tilt of the CMB power spectra, degeneracies withns and Ωbh2 are expected. Indeed, we find the errors of these param-eters to be slightly increased, while all other parameters are basi-cally unaffected (see Fig. 13). In particular, the θMC contours still

mimic the Planck 2015 contours without varying αEM as shown bythe red and dotted black line in Fig. 13. This already indicates thatthe individual effects of variations of αEM(z0) and p should be sepa-rable. When varying both parameters independently, we obtain theconstraints indicated by the last column in Table 1. Albeit slightlyweakened, we can independently constrain αEM(z0) and p.

Carrying out a similar analysis for me, setting me(z0) = me,0 wefind p = 0.0006±0.0044 for Planck 2015 TTT EEE+lowP+lensingdata. This is roughly 2 times weaker than for αEM, consistent withnaive analysis of the free electron fraction scaling around z ' 1100(see Fig. 5). Also varying me(z0), we obtain me/me,0 = 0.960+0.046

−0.071and p = 0.0012+0.0047

−0.0042. When adding BAO data, this improves tome/me,0 = 1.0023 ± 0.0074 and p = 0.0007 ± 0.0043, again withno significant biases in the standard parameters with respect to thePlanck 2015 cosmology remaining.

5 CONCLUSIONS

Current observations provide us with very precise cosmologicaldatasets, that allow us to ask detailed questions about the condi-tions of the Universe around the recombination epoch. In this pa-per, we analyzed the different effects on the recombination problemwhen varying αEM and me. We explained the modifications to therecombination codes, Recfast++ and CosmoRec, that are requiredto vary these constants in an easy and efficient way. In particu-lar, we developed an improved correction function treatment forRecfast++ (Sect. 2.1.1), which allows us to accurately representthe full computation of CosmoRec (cf. Fig. 5). We find that the re-maining differences between the two recombination codes can inprinciple be neglected at the current level of precision.

For constant |∆α/α| . 1%, we find a total effect on the ioniza-tion history of ∆Xe/Xe ' −27 × ∆α/α at z ' 103 (see Fig. 3). Thisis dominated by the required rescaling of the ionization potential inthe equilibrium Boltzmann factors. Other corrections related to A2γ,αrec and βphot contribute at the ∼ 10% level to this net effect. We alsofind that when varying αEM and me the associated direct changes tothe recombination history caused by scaling the Thomson scatter-ing cross section are negligible (see Fig. 3). We still include thiscorrection in our analysis for consistency.

When varying me, we find the net effect on Xe around z ' 103

to be comparable to that of varying αEM for ∆me/me ≈ 2.5×∆α/α.The net change of Xe in the freeze-out tail is smaller than for αEM

(see Fig. 4 and 5), an effect that is related to the different scaling ofαrec and βphot with αEM and me (see Sect. 2.2).

We also include explicit redshift-dependent variations ofαEM. This has a very different effect on the ionization historyaround recombination. Instead of shifting the recombination red-shift during hydrogen recombination, the ionization history isstretch/compressed differentially, depending on the chosen param-eters (see Fig. 6). This has a distinct effect on the CMB anisotropiesthat can be separated from the one for constant variations.

The propagation of the modifications in the recombinationdynamics through to the Thomson visibility function and CMBanisotropies is also illustrated (see Sect. 3). For constant ∆α/α, ourresults are in agreement with previous analyses (e.g., Kaplinghatet al. 1999; Battye et al. 2001). We find that the changes to the CMBtemperature power spectrum caused by variation of αEM and me arepractically degenerate when ∆me/me ≈ (2 − 3) × ∆α/α ' 10−3

(see Fig. 10). However, combined constraints on αEM and me areobtained in a regime in which higher order terms especially for me

become relevant (Sect. 4.1.1), making them again distinguishable.

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Variation of αEM and me 11

Changes caused by directly varying the CMB monopole tempera-ture, T0, should in principle be distinguishable due to the ISW ef-fects (see Sect. 3.1.4 and Fig. 10), although we do not explore thispossibility in more detail here.

We also illustrate the effect of redshift-dependent changes. In-stead of shifting the maximum of the Thomson visibility function(cf. Fig. 7), a power-law variation of αEM with redshift (see Eq. 4)causes a change in the width of the visibility function (see Fig. 11).This primarily modifies the blurring of CMB anisotropies (compareFig. 8 and 12) and can thus be distinguished.

In Sect. 4, we present our constraints on different cases us-ing Planck 2015 data. Our results (see Table 1 and 2) for constant∆α/α and ∆me/me are consistent with those given in Planck Col-laboration et al. (2015b). We obtain the updated individual con-straints αEM/αEM,0 = 0.9993 ± 0.0025 and me/me,0 = 0.961+0.046

−0.072using Planck 2015 data alone. Also adding BAO data, we findαEM/αEM,0 = 0.9997 ± 0.0023 and me/me,0 = 1.0039 ± 0.0074.When varying me, the addition of BAO data removes the bias inH0 towards low values, making the results consistent with the stan-dard Planck cosmology (Table 2). Simultaneous constraints whenvarying both αEM and me are presented in Sect. 4.1.1.

Although we show that the effect of rescaling σT for the com-putation of the Thomson visibility function is quite small (cf., Fig. 7and 11), this effect should be included in the analysis (see alsoPlanck Collaboration et al. 2015b). For models with varying αEM,we find that this effect improves the constraint by ' 30%. Thechange is more dramatic when varying me. Here, we find that ne-glecting the rescaling of σT leads to a significant underestimationof the error (a factor & 5) unless BAO data is added. This is dueto enhanced geometric degeneracies caused by the scaling of σT inthe visibility function calculation (see Sect. 4.1).

Allowing for power-law redshift dependence of αEM aroundz0 = 1100 with αEM(z0) = αEM,0, we find the new constraint,p = 0.0008 ± 0.0025, on the power-law index. When varyingboth αEM(z0) and p, we obtain αEM(z0)/αEM,0 = 0.9998 ± 0.0036and p = 0.0006 ± 0.0036 (see Table 1). Similarly, for me we findp = 0.0006 ± 0.0044 (CMB only) assuming me(z0) = me,0. Vary-ing both me(z0) and p we obtain me/me,0 = 1.0023 ± 0.0074 andp = 0.0007 ± 0.0043 when also adding BAO data (see Sect 4.2).All these results are fully consistent with the standard values, high-lighting the impressive precision, stability and consistency of thedata with respect to non-standard extensions. This also suggeststhat a wider class of varying fundamental constant models can inprinciple be probed using the CMB, possibly with more complexredshift-dependence (e.g., phase transition, spikes or higher ordertemporal curvature).

Modified recombination physics can also be investigated usingCMB spectral distortions. For the future, we aim to continue thisstudy with the cosmological recombination radiation (e.g., Rubino-Martın et al. 2006; Chluba & Sunyaev 2006a; Sunyaev & Chluba2009). Modeling these variations in CosmoSpec (Chluba & Ali-Haımoud 2016) will enlighten us on how the fundamental constantschange the recombination spectrum and provide us with anotherdataset for constraints. This could allow us to alleviate existing pa-rameter degeneracies and further deepen our understanding of therecombination epoch, allowing us to confront clear theoretical pre-dictions with direct observational evidence. This might also openthe possibility to probe the redshift-dependence of the fundamen-tal constants at even earlier phases through the individual effectson hydrogen and helium recombination (e.g., see Fig. 6 for the ef-fect on Xe), which would remain inaccessible otherwise. One could

furthermore refine constraints on spatial variations of fundamentalconstants. We look forward to exploring these opportunities.

ACKNOWLEDGEMENTS

We cordially thank the referee for their comments on the paper and the sug-gestions to more carefully consider the effect of σT and extend our analysisfor me. We also thank Marcos Ibanez from IAC for his discussion surround-ing CosmoMC and how to optimise the sampling, given an increase in param-eters. We extend thanks to Richard Battye, Carlos Martins and James Richfor discussion pertaining to previous studies of fundamental constant varia-tions. We would also like to thank the Jodrell Bank Centre for Astrophysics,University of Manchester for the use of the Fornax cluster as a tool to helpreduce convergence runtimes. LH is funded by the Royal Society throughgrant RG140523. JC is supported by the Royal Society as a Royal SocietyUniversity Research Fellow at the University of Manchester, UK.

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